Theorem fliftf 5778

Description: The domain and range of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftf	|- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftf

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 |- ( ( ph /\ Fun F ) -> Fun F )
2		flift.1	. . . . . . . . . . 11 |- F = ran ( x e. X |-> <. A , B >. )
3		flift.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> A e. R )
4		flift.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> B e. S )
5	2, 3, 4	fliftel 5772	. . . . . . . . . 10 |- ( ph -> ( y F z <-> E. x e. X ( y = A /\ z = B ) ) )
6	5	exbidv 1818	. . . . . . . . 9 |- ( ph -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
7	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
8		rexcom4 2753	. . . . . . . . 9 |- ( E. x e. X E. z ( y = A /\ z = B ) <-> E. z E. x e. X ( y = A /\ z = B ) )
9		19.42v 1899	. . . . . . . . . . . 12 |- ( E. z ( y = A /\ z = B ) <-> ( y = A /\ E. z z = B ) )
10		elisset 2744	. . . . . . . . . . . . . 14 |- ( B e. S -> E. z z = B )
11	4, 10	syl 14	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> E. z z = B )
12	11	biantrud 302	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( y = A <-> ( y = A /\ E. z z = B ) ) )
13	9, 12	bitr4id 198	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( E. z ( y = A /\ z = B ) <-> y = A ) )
14	13	rexbidva 2467	. . . . . . . . . 10 |- ( ph -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
15	14	adantr 274	. . . . . . . . 9 |- ( ( ph /\ Fun F ) -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
16	8, 15	bitr3id 193	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z E. x e. X ( y = A /\ z = B ) <-> E. x e. X y = A ) )
17	7, 16	bitrd 187	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. x e. X y = A ) )
18	17	abbidv 2288	. . . . . 6 |- ( ( ph /\ Fun F ) -> { y | E. z y F z } = { y | E. x e. X y = A } )
19		df-dm 4621	. . . . . 6 |- dom F = { y | E. z y F z }
20		eqid 2170	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
21	20	rnmpt 4859	. . . . . 6 |- ran ( x e. X |-> A ) = { y | E. x e. X y = A }
22	18, 19, 21	3eqtr4g 2228	. . . . 5 |- ( ( ph /\ Fun F ) -> dom F = ran ( x e. X |-> A ) )
23		df-fn 5201	. . . . 5 |- ( F Fn ran ( x e. X |-> A ) <-> ( Fun F /\ dom F = ran ( x e. X |-> A ) ) )
24	1, 22, 23	sylanbrc 415	. . . 4 |- ( ( ph /\ Fun F ) -> F Fn ran ( x e. X |-> A ) )
25	2, 3, 4	fliftrel 5771	. . . . . . 7 |- ( ph -> F C_ ( R X. S ) )
26	25	adantr 274	. . . . . 6 |- ( ( ph /\ Fun F ) -> F C_ ( R X. S ) )
27		rnss 4841	. . . . . 6 |- ( F C_ ( R X. S ) -> ran F C_ ran ( R X. S ) )
28	26, 27	syl 14	. . . . 5 |- ( ( ph /\ Fun F ) -> ran F C_ ran ( R X. S ) )
29		rnxpss 5042	. . . . 5 |- ran ( R X. S ) C_ S
30	28, 29	sstrdi 3159	. . . 4 |- ( ( ph /\ Fun F ) -> ran F C_ S )
31		df-f 5202	. . . 4 |- ( F : ran ( x e. X |-> A ) --> S <-> ( F Fn ran ( x e. X |-> A ) /\ ran F C_ S ) )
32	24, 30, 31	sylanbrc 415	. . 3 |- ( ( ph /\ Fun F ) -> F : ran ( x e. X |-> A ) --> S )
33	32	ex 114	. 2 |- ( ph -> ( Fun F -> F : ran ( x e. X |-> A ) --> S ) )
34		ffun 5350	. 2 |- ( F : ran ( x e. X |-> A ) --> S -> Fun F )
35	33, 34	impbid1 141	1 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   C_ wss 3121   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   X. cxp 4609   dom cdm 4611   ran crn 4612   Fun wfun 5192   Fn wfn 5193   -->wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  qliftf  6598